The HS equation also describes the high-frequency limit [1]}, [2]} of the Camassa-Holm (CH) equation
–a model equation for shallow
water waves [3]}, [4]}, [5]}
and a re-expression of the geodesic flow on the diffeomorphism group of
the circle [6]} with a bi-Hamiltonian structure [7]} which is completely integrable [8]}.
The HS equation is also a completely integrable system with a bi-Hamiltonian structure, and hence, it possesses
a Lax pair, an infinite family of commuting Hamiltonian flows, as well as an associated sequence of
conservation laws [9]}, [10]}, [2]}, [12]}, [13]}. It also describes
the geodesic flow on the homogeneous space related to the Virasoro group [14]}, [15]}, [16]}.
Recently, algebro-geometric solutions for the HS hierarchy was investigated in [17]}.
Moreover, peakon solutions, global weak solutions and the Cauchy problem of the system
(REF ) were discussed in [18]}, [19]}, [20]}, [21]}.
However, within the knowledge of the authors, the
algebro-geometric solutions of the entire HS2 hierarchy are not studied yet.
